Spontaneous baryosynthesis with large initial phase

Technical Summary: Spontaneous Baryosynthesis with Large Initial Phase

Problem Statement
The observed baryon asymmetry of the universe, quantified by the ratio $(\Delta n_B/s)_0 \simeq 8.6 \times 10^{-11}$, remains a fundamental challenge in cosmology. While spontaneous baryogenesis offers a mechanism where a pseudo-Nambu-Goldstone boson (pNGB) generates asymmetry through the relaxation of a misalignment angle $\theta$, standard analyses rely heavily on the small-angle approximation ($\theta_i \ll 1$). However, cosmological inflation induces quantum fluctuations in the pNGB field, leading to a Gaussian probability distribution for the initial misalignment angle $\theta_i$. Statistical analysis suggests that while the probability of a single Hubble patch having $\theta_i \sim \pi$ is low, the vast number of independent patches ($\sim e^{180}$) makes the existence of such regions within the observable universe statistically certain. The problem addressed is whether the established cubic scaling of baryon asymmetry with the initial phase holds for these large misalignment angles, particularly near the potential maximum at $\theta_i = \pi$.

Methodology
The authors investigate the dynamics of the pNGB field $\theta$ within the framework of spontaneous baryogenesis, assuming the field acts as a spectator during inflation. The theoretical model involves a complex scalar field $\Phi$ undergoing spontaneous symmetry breaking, coupled to heavy fermions $Q$ and $L$ via a Yukawa interaction. An explicit symmetry-breaking potential $V(\theta) = \Lambda^4(1 - \cos \theta)$ is introduced to give the pNGB a mass and drive its evolution.

The study proceeds in two main stages:

1. Probability Distribution Analysis: The authors derive the probability distribution $f(\theta_i)$ for the initial misalignment angle at the end of inflation using the Fokker-Planck equation for a massless scalar field. They calculate the likelihood of regions existing where $\theta_i$ deviates significantly from the mean, specifically targeting the regime near $\pi$.
2. Numerical Simulation in Minkowski Spacetime: To isolate the effects of large initial phases without the complications of cosmic expansion, the authors solve the semiclassical equation of motion for $\theta$ in Minkowski space. They treat the fermion fields quantum mechanically while keeping the scalar field classical. The equation of motion includes a non-local memory integral representing particle production. To make this numerically tractable, the authors introduce a physical cutoff $\omega \sim f$ (the symmetry-breaking scale) and approximate the memory integral to derive a local equation of motion with an effective damping term:
   $$\ddot{\theta} + \Gamma_{\text{eff}}\dot{\theta} + \frac{\Lambda^4}{f^2}\sin\theta = 0$$
   Here, $\Gamma$ is treated as a phenomenological parameter. The authors numerically solve this differential equation for various damping rates ($\Gamma$) and initial conditions, specifically starting from $\theta_i \approx \pi$ with $\dot{\theta}_i = 0$. Finally, they compute the baryon and antibaryon number densities using the time-dependent solutions of $\theta(t)$ to determine the net asymmetry.

Key Contributions and Results

• Validation of Small-Angle Regime: The numerical method successfully reproduces the established result from Ref. [8] that for small initial phases, the baryon asymmetry scales cubically with the initial angle ($\Delta n_B \propto \theta_i^3$).
• Breakdown of Cubic Scaling: For large initial angles, the cubic dependence breaks down. The analysis reveals that as $\theta_i$ approaches $\pi$, the production of particles does not continue to increase indefinitely but rather saturates.
• Dynamics Near $\pi$: The authors observe that for large initial phases, the oscillation period of the field becomes significantly longer than that predicted by the harmonic approximation ($T \sim 1/m_\theta$). This extended period affects the efficiency of particle production.
• Impact of Damping: The study explores the dependence on the damping rate $\Gamma$. While small $\Gamma$ values show a specific saturation behavior, larger $\Gamma$ values (stronger damping) result in a more pronounced deviation from the small-angle behavior, though the asymmetry remains of the same order of magnitude.
• Minkowski Approximation: The authors note that their results are specific to Minkowski spacetime, valid when the decay rate of the pNGB oscillations is significantly greater than the Hubble expansion rate.

Significance and Claims
The paper claims to provide a necessary correction to the standard treatment of spontaneous baryogenesis by moving beyond the convenient but potentially limiting small-angle approximation. By demonstrating that regions with $\theta_i \sim \pi$ are statistically inevitable in the early universe, the authors argue that the mechanism's behavior in this regime must be understood.

The primary finding is that while the small-angle approximation predicts a specific scaling, the physics of large misalignment angles leads to a saturation of particle production as the initial phase approaches the potential maximum. The authors conclude that, within the context of Minkowski spacetime, the effects of large misalignment angles on the generated baryon asymmetry are not "substantially different" in terms of order of magnitude from the small-angle predictions, despite the change in functional dependence. The work serves as a foundational numerical investigation, establishing that the mechanism remains viable even for extreme initial conditions, though the detailed dynamics differ from the perturbative regime.